We study smooth classes without the algebraic closure property. For such smooth classes we investigate the simplicity of the class of generic structures, in the context of Robinson theories.

Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable 0-categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwin's almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In (...) particular, the generic structure need not be ω-saturated and so the argument for stability is significantly more complicated. We further show that these structures are “flat” and do not interpret a group. (shrink)

We construct a new class of 1 categorical structures, disproving Zilber's conjecture, and study some of their properties.